Robust Asset Allocation

R. H. Tutuncu∗ and M. Koenig†

August 20, 2002Revised September 18, 2003

Abstract

This article addresses the problem of finding an optimal allocation of funds amongdifferent asset classes in a robust manner when the estimates of the structure of returnsare unreliable. Instead of point estimates used in classical mean-variance optimization,moments of returns are described using uncertainty sets that contain all, or most, oftheir possible realizations. The approach presented here takes a conservative viewpointand identifies asset mixes that have the best worst-case behavior. Techniques forgenerating uncertainty sets from historical data are discussed and numerical resultsthat illustrate the stability of robust optimal asset mixes are reported.

Key words: Robust optimization, mean-variance optimization, saddle-point prob-lems.

1 Introduction

Portfolio optimization is one of the best known and most widely used methods infinancial portfolio selection. Developed by Harry Markowitz (1952) five decades ago,this approach quantifies the trade-off between the expected return and the risk ofportfolios of financial securities using mathematical techniques and offers a method fordetermining a frontier of optimal (Pareto-efficient) portfolios. Since risk is measuredby the variance of the random portfolio return in this approach, it is also called mean-variance optimization (MVO). Here, we address asset allocation problems, i.e., theproblem of finding an optimal allocation of funds among different asset classes whichcan be formulated in an identical manner to MVO.

Often, the set of optimal or efficient portfolios is described using a two-dimensionalgraph called the efficient frontier that plots their expected returns and standard de-viations. Each efficient portfolio can be identified by solving an associated convex

∗Associate Professor, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh,Pennsylvania.

†Director of Quantitative Analysis, National City Investment Management Company, Cleveland, Ohio.

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quadratic programming (QP) problem, a well-studied problem in the optimization lit-erature. To determine the entire efficient frontier ranging from the portfolio with thesmallest overall variance to the portfolio with the highest expected return, one has tosolve a parametric QP problem using, for example, Markowitz’ method of critical lines.

Despite the elegance of the model developed by Markowitz, the powerful optimiza-tion theory supporting this model, and the availability of efficient software to solve theresulting problem, MVO continues to encounter skepticism among investment practi-tioners. One reason usually cited for this skepticism is the counter-intuitive nature ofthe optimal portfolios generated by the MVO approach. Optimal portfolios tend toconcentrate on a small subset of the available securities, and appear not to be welldiversified. Furthermore, optimal portfolios are often sensitive to changes in the in-put parameters of the problem (expected returns and the covariance matrix) and leadto large turnover ratios with periodic readjustments of the input estimates; see forexample Michaud (1989, 1998).

This last observation indicates that the inputs to the MVO model need to be veryaccurately estimated. However, this is a very difficult task, especially in the case ofexpected return estimates. Different techniques used in moment estimation can anddo generate significantly different point estimates of MVO inputs which, in turn, leadto large variations in the composition of efficient portfolios. Using estimates from aparticular source in the MVO model introduces an estimation risk in portfolio choice,and methods for optimal selection of portfolios must take this risk into account, seeBawa, Brown, and Klein (1979).

Robust optimization, an emerging branch of the field of optimization, offers vehiclesto incorporate estimation risk into the decision making process in portfolio choice/assetallocation. Generally speaking, robust optimization refers to finding solutions to givenoptimization problems with uncertain input parameters that will achieve good objec-tive values for all, or most, realizations of the uncertain input parameters. It shouldbe noted, however, that there are different interpretations of robustness that lead todifferent mathematical formulations–see Jen (2001) for at least 17 different definitionsof robustness in different contexts. Here, we take the pessimistic view of robustnessand look for a solution that has the best performance under its worst case.

In our approach, uncertainty is described using an uncertainty set which includesall, or most, possible realizations of the uncertain input parameters. Given a problemwith uncertain inputs and an uncertainty set for these inputs, our robust optimizationapproach addresses the following problem: What choice of the variables of the problemwill optimize the worst case objective value? That is, for each choice of the decisionvariables, we consider the worst case realization of the data and evaluate the corre-sponding objective value, and then pick the set of values for the variables with thebest worst-case objective. We apply this approach to the portfolio selection problemusing a judicious choice of the uncertainty set. We demonstrate that the resultingrobust optimization problem is simple in some cases meaning that it can be solvedas a standard quadratic programming problem. In most cases, however, this simplifi-cation is not possible. For such cases we formulate the robust optimization problemas a saddle-point problem and apply an interior-point algorithm to this saddle-pointproblem.

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While the approach in this article is related to the methods in Lobo et al. (1999),Goldfarb and Iyengar (2003), the formulation and the algorithm used here are basedon those developed by Halldorsson and Tutuncu (2003). In addition to presentinga variation of their formulation and discussing a new formulation for identifying ro-bust portfolios with the largest Sharpe ratio, we discuss an implementation of thesealgorithms.

We also address the issue of generating uncertainty sets and describe two ap-proaches; one based on bootstrapping and the other on moving averages. Since ourworst-case based approach can be detrimentally influenced by outliers in the data, un-certainty sets need to be carefully chosen. This is why we may prefer to include mostrather than all possible realizations of the uncertain parameters in uncertainty sets.To minimize outlier effects, we eliminate some of the lowest and highest quantiles ofthe processed data in both the bootstrapping and moving averages strategies and usethe remaining data to define uncertainty sets.

Our numerical experiments indicate that robust asset allocation is indeed a valu-able alternative for conservative investors. Robustness is achieved at relatively littlecost–robust efficient portfolios are only marginally inefficient when faced with nominalinputs. In contrast, efficient portfolios derived from nominal inputs can be severely in-efficient under worst-case realizations of the uncertain parameters. We further demon-strate that robust optimal allocations are stable in the sense that re-solving the robustasset allocation problem periodically as new data is collected results in essentially un-changed portfolios. This type of low turnover is often attractive for long-term investors.

The remainder of this article is organized as follows: Section 2 presents formulationsof problems to find robust optimal allocation of assets and robust portfolios with themaximum Sharpe ratio. In Section 3, we present a rigorous description of the methodwe implemented to determine the robust efficient frontier. A detailed description of akey subroutine is given in the Appendix. Numerical experiments and their results arediscussed in Section 4. We present our conclusions in Section 5.

2 Robust Optimization Problems

2.1 The Robust MVO Problem

Optimal portfolio selection/asset allocation problems can be formulated mathemati-cally as quadratic programming (QP) problems. Convex QP refers to minimizing aconvex quadratic function (or, equivalently, maximizing a concave quadratic function)subject to linear equality and inequality constraints. Solution of a convex QP associ-ated with an asset allocation problem generates an efficient portfolio on the efficientfrontier. To generate the entire efficient frontier, the QP has to be parametrized andthis can be done in three essentially equivalent ways: (i) maximize expected returnsubject to an upper limit on the variance, (ii) minimize the variance subject to a lowerlimit on the expected return, (iii) maximize the risk-adjusted expected return. Thesethree problems are parametrized by the variance limit, expected return limit, and therisk-aversion parameter, respectively. Since the variance constraint is nonlinear, the

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first formulation is not a QP. Hence, we focus on the last two of these formulations:

minx∈<n xTQxs.t. µTx ≥ R

x ∈ X, (1)

maxx∈<n µTx− λxTQxs.t. x ∈ X . (2)

Above, µi–the ith component of the vector µ–denotes the estimated expected returnof security i. The matrix Q is the covariance matrix of these returns. Diagonal elementsqii of the Q matrix denote the variance of the return on security i while off-diagonalelements qij denote the covariance between the returns of securities i and j. Thecomponents xi of the variable vector x denote the proportion of the portfolio to beinvested in security i. R in the right-hand-side of (1) is the lower limit on the expectedreturn one would like to achieve. X represents the polyhedral set of feasible portfolios,i.e., portfolios that satisfy certain linear constraints imposed by the investor. Thescalar λ in (2) represents the risk-aversion parameter mentioned above. The objectivefunction of problem (2) represents a risk-adjusted expected return function. Since thecovariance matrix Q is always positive semidefinite, problems (1) and (2) are convexquadratic programming problems solvable in polynomial time.

By solving (1) and (2) for different values of R and λ, one can generate a sequence ofoptimal portfolios on the efficient frontier ranging from the portfolio with the smallestoverall variance to the portfolio with the highest expected return. Problems (1) and(2) are equivalent in the following sense: If x∗(λ) solves (2) for a fixed value of λ, italso solves (1) for R = µTx∗(λ).

We will make the reasonable assumption that the set X of feasible portfolios isnon-empty and bounded. For example,

X = {x ∈ <n|n∑

i=1

xi = 1, x ≥ 0} (3)

corresponds to the case where short-sales are not allowed and satisfies this assumption.In a more general setting, X may contain additional constraints corresponding to sectordistribution requirements, minimal investment restrictions, bounds on the total numberof assets in the portfolio, etc.

We argued above that one of the main criticisms against the MVO approach centerson the observation that the optimal portfolios generated by this approach are oftenquite sensitive to the input parameters–µ and Q in our notation. To make mattersworse, these parameters can never be observed and one has to settle for estimatesfound using some particular technique. In the presence of, say, equally reliable multipleestimates for µ and Q, the best way to integrate all available information is no longerclear. Our strategy is to represent all the available information on the unknown inputparameters in the form of an uncertainty set, i.e., a set that contains most of thepossible values for these parameters.

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In the case of the MVO problems (1) and (2), an uncertainty set for the expectedreturn vector µ and the covariance matrix Q may take the form of intervals:

Uµ = {µ : µL ≤ µ ≤ µU}, (4)UQ = {Q : QL ≤ Q ≤ QU , Q � 0}, (5)U = {(µ,Q) : µ ∈ Uµ, Q ∈ UQ}. (6)

Above, µL, µU , QL, QU are the extreme values of the intervals we just mentioned. Therestriction Q � 0 indicates that Q is a symmetric positive semidefinite matrix which isa necessary property for the uncertain Q to be a covariance matrix. The uncertaintyset U need not be separable as above, but we focus on separable sets in this paper asthey appear more natural than compound uncertainty sets U that are not separable.

There are, of course, other ways to describe uncertainty sets. For example, theycould be discrete sets representing a collection of estimates for the unknown input pa-rameters. Goldfarb and Iyengar (2003) consider ellipsoidal uncertainty sets. We preferintervals and there are several reasons for this preference. For example, the end-pointsof the interval may correspond to the extreme values of the corresponding statistic inhistorical data, in analyst estimates, in simulated scenarios, etc. Alternatively, a mod-eler may choose a confidence level and then generate estimates of return and covarianceparameters in the form of prediction intervals. For the examples we present in Section4, we generated the bounds µL, µU , QL, QU using percentiles of bootstrapped samplesof historical data as well as the percentiles of moving averages.

Given the uncertainty set U , the robust versions of the mean-variance optimizationproblems (1) and (2) can be expressed as follows:

min {maxQ∈UQxTQx}

s.t. x ∈ Xminµ∈Uµ µ

Tx ≥ R,(7)

maxx∈X

{ minµ∈Uµ,Q∈UQ

µTx− λxTQx}. (8)

The minimax problem in (7) was formulated by Goldfarb and Iyengar (2003). Themaximin problem in (8) was formulated in Halldorsson and Tutuncu (2003) where asaddle-point representation and a solution algorithm were also provided. We discusstheir algorithm in the Appendix.

We have argued above that the problems (1) and (2) are equivalent. There is asimilar equivalence between the robust optimization problems (7) and (8) as well:

Proposition 1 Let x∗(λ) denote an optimal solution of (8) for a given positive valueof λ. Then, x∗(λ) is also an optimal solution of (7) for

R = minµ∈Uµ

µTx∗(λ). (9)

Proof:Since objective function of the inner minimization problem in (8) is separable, this prob-lem can be solved by solving the two smaller problems minµ∈Uµ µ

Tx and maxQ∈UQxTQx.

Let µ∗(λ) and Q∗(λ) be the optimal solutions to these problems corresponding to x∗(λ).

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Now, if x∗(λ) is not optimal for (7) with R as in (9), there must exist x ∈ X andQ ∈ UQ satisfying the following:

minµ∈Uµ

µT x ≥ R = minµ∈Uµ

µTx∗(λ) = µ∗(λ)Tx∗(λ),

maxQ∈UQ

xTQx = xT Qx < x∗(λ)TQ∗(λ)x∗(λ).

But this would imply that

min(µ,Q)∈U

(µT x− λxTQx

)= min

µ∈Uµ

µT x− λxT Qx > µ∗(λ)Tx∗(λ)− λx∗(λ)TQ∗(λ)x∗(λ)

contradicting the optimality of x∗(λ) for (8). Thus, x∗(λ) must be optimal for (7) aswell when R = minµ∈Uµ µ

Tx∗(λ).

2.2 Robust Portfolios with the Maximum Sharpe Ratio

There are assets such as US Treasury Bills that can be considered essentially risklessfor investment or borrowing purposes. If such an asset is included in the asset pool foroptimal portfolio selection, all efficient portfolios turn out to be linear combinations ofthis riskless asset and a unique optimal risky portfolio. This optimal risky portfolio isthe efficient portfolio with the highest Sharpe ratio, which is defined as follows for aportfolio x:

h(x) =µTx− rf√xTQx

. (10)

Here, rf represents the known return on the riskless asset. This quantity represents thereward to variability (measured by the standard deviation) ratio of a zero-investmentportfolio funded by riskless borrowing. We assume that there are feasible portfolioswith positive Sharpe ratios–otherwise, the riskless asset is the only efficient portfolio.

In this section, we assume that Q is positive definite so that xTQx > 0 and,therefore, h(x) is defined for all non-zero portfolios. Since the covariance matrix Qis always positive semidefinite, this assumption is equivalent to assuming that Q isnonsingular and is essentially equivalent to assuming that there are no redundantassets in our collection.

The portfolio with the highest Sharpe ratio is obtained by solving the optimizationproblem

maxh(x) s.t. x ∈ X . (11)

This maximization problem, however, has a nonlinear and non-concave objective func-tion and may be difficult to solve directly. In what follows, we generalize an approachintroduced by Goldfarb and Iyengar (2003), to reduce this problem into a convex mini-mization problem, which is easier to solve. Then, we will present the robust formulationcorresponding to this problem.

The elegant argument in Goldfarb and Iyengar (2003) is based on the simple ob-servation that eTx = 1 whenever x ∈ X (e represents an n-dimensional vector of 1’s)

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since proportions in all securities must sum to 1, and therefore, h(x) can be rewrittenas a homogeneous function of x–we call this function g(x):

h(x) =µTx− rf√xTQx

=(µ− rfe)Tx√

xTQx=: g(x) = g(

x

κ), ∀κ > 0.

The vector µ− rfe is the vector of returns in excess of the risk-free lending rate. Gold-farb and Iyengar demonstrate that when X has the form in (3), using the argumentabove, one can replace the normalization constraint eTx = 1 with the alternative nor-malization constraint (µ−rfe)Tx = 1 without affecting the optimal solution. But then,the objective function is equivalent to minimizing xTQx, a strictly convex quadraticfunction of x (recall our assumption that Q is a positive definite matrix).

We show that, a similar reduction can be achieved even when X is not in the formin (3), as long as x ∈ X implies that eTx = 1. To achieve the desired reduction, wefirst homogenize X applying the lifting technique to it, i.e., we consider a set X+ thatlives in a one higher dimensional space than X and is defined as follows:

X+ := {x ∈ <n, κ ∈ <|κ > 0,x

κ∈ X} ∪ (0, 0). (12)

We add the vector (0, 0) to the set to achieve a closed set. Note that X+ is a cone. Forexample, when X is a circle, X+ resembles an ice-cream cone. When X is polyhedral,e.g., X = {x|Ax ≥ b, Cx = d}, we have X+ = {(x, κ)|Ax−bκ ≥ 0, Cx−dκ = 0, κ ≥ 0}.Now, using the observation that h(x) = g(x),∀x ∈ X and that g(x) is homogeneous,we conclude that (11) is equivalent to

max g(x) s.t. (x, κ) ∈ X+. (13)

Again, using the observation that g(x) is homogeneous in x, we see that adding thenormalizing constraint (µ−rfe)Tx = 1 to (13) does not affect the optimal solution–fromamong a ray of optimal solutions, we will find the one on the normalizing hyperplane.Note that for any x ∈ X with (µ − rfe)Tx > 0, the normalizing hyperplane willintersect with an (x+, κ+) ∈ X+ such that x = x+/κ+. In fact, x+ = x

(µ−rf e)T xand

κ+ = 1(µ−rf e)T x

. The normalizing hyperplane will miss the rays corresponding to points

in X with (µ− rfe)Tx ≤ 0, but since they can not be optimal, this will not affect theoptimal solution. Therefore, substituting (µ − rfe)Tx = 1 into g(x) we obtain thefollowing equivalent problem:

max1√xTQx

s.t. (x, κ) ∈ X+, (µ− rfe)Tx = 1. (14)

Thus, we proved the following result:

Proposition 2 Given a set X of feasible portfolios with the property that eTx =1, ∀x ∈ X , the portfolio x∗ with the maximum Sharpe ratio in this set can be found bysolving the following problem with a convex quadratic objective function

minxTQx s.t. (x, κ) ∈ X+, (µ− rfe)Tx = 1, (15)

with X+ as in (12). If (x, κ) is the solution to (15), then x∗ = xκ .

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As in Goldfarb and Iyengar (2003), we observe that the normalizing constraint in (15)can be relaxed to (µ − rfe)Tx ≥ 1 by recognizing that this constraint will always betight at an optimal solution. The relaxed problem will be in the form (1) and therefore,its robust version can be formulated as follows:

min {maxQ∈UQxTQx}

s.t. (x, κ) ∈ X+

minµ∈Uµ(µ− rfe)Tx ≥ 1.(16)

3 Finding Robust Portfolios

In this section, we discuss methods for solving the robust formulation (8) of the MVOproblem presented in the introduction. First, we discuss a special case of the robustoptimization formulation that can be solved as a standard QP problem and therefore,does not require the development of any new solution techniques:

3.1 The Simple Case

In most asset allocation problems, short sales are not allowed. Money managers oftenlook for a nonnegative portfolio of mutual funds representing different asset classes. Ifthere are no additional considerations for the asset allocation problem, the feasible setof portfolios has precisely the description given in (3):

X = {x ∈ <n|n∑

i=1

xi = 1, x ≥ 0}.

Above, x ≥ 0 represent the “no-short-sales” constraint and the restriction∑n

i=1 xi = 1is necessary to ensure that all the money available for investment is allocated.

Now, we consider an uncertainty set U of the form (6) with the property that thematrix QU is positive semidefinite. In this case, we have the following result thatsimplifies the search for robust portfolios:

Proposition 3 Let x ∈ <n be a nonnegative vector and let U be as in (4)–(6) with apositive semidefinite matrix QU . Then, an optimal solution of the problem

min(µ,Q)∈U

µTx− λxTQx (17)

is µ∗ = µL and Q∗ = QU regardless of the values of the nonnegative scalar λ and thevector x.

Proof:First note that both the objective function µTx− λxTQx and the constraint set U areseparable in µ and Q. Therefore, as in the proof of Proposition 1, the problem (17)can be solved by solving the following two smaller problems:

min µTxs.t. µL ≤ µ ≤ µU ,

max xTQxs.t. QL ≤ Q ≤ QU

Q � 0. (18)

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Since x ≥ 0, the objective value of the first problem in (18) is minimized when eachelement of the vector µ is at its lower bound, i.e., when µ = µL. Consider the relaxationof the second problem obtained by ignoring the positive semi-definiteness constraintQ � 0. For this relaxation, since xixj ≥ 0 for all i and j, xTQx =

∑i,j qijxixj will

be maximized when all qij attain their largest feasible values, i.e., when Q = QU .Since QU is assumed to be a positive semidefinite matrix, it must be optimal for theunrelaxed problem as well.

The proposition above indicates that when short sales are not allowed and whenthe upper bounds on the covariance matrix form an acceptable covariance matrix, thenthe worst-case realization of the data is the same regardless of what portfolio is chosen–expected returns are realized at their lowest possible values and the covariances arerealized at their highest possible values. In this scenario, the maximin problem givenin (8) reduces to the following maximization problem:

maxx∈X

(µL

)Tx− λxTQUx. (19)

This is a standard MVO problem and the associated efficient frontier can be determinedusing methods such as the method of critical lines in Markowitz (1952). A similarargument shows that the minimax problem (7) reduces to the following minimizationproblem when QU is positive semidefinite:

min xTQUxs.t. x ∈ X(

µL)T

x ≥ R.

(20)

3.2 The General Case

In the previous subsection we saw that the robust asset allocation problem can bereduced to a simple MVO problem under certain assumptions. When these assumptionsdo not hold, the worst-case realization of the uncertain inputs is no longer the samefor all possible portfolios. This means that we cannot expect to solve the robust assetallocation problem in a sequential manner, i.e., by first finding the worst-case inputdata and then finding the best allocation with this data, as we did in the previoussubsection. Fortunately, the robust problem can still be solved using a saddle-pointformulation as we describe below. The arguments below follow the construction inHalldorsson and Tutuncu (2003):

First we need to introduce some notation to represent the objective function of therobust optimization problem (8). Let

ψλ(x, µ,Q) := µTx− λxTQx, x ∈ X , (µ,Q) ∈ U . (21)

For fixed (µ,Q) ∈ U and a given λ ≥ 0, the function ψλ is a concave quadratic functionof x. Similarly, for fixed x and a given λ, the function ψλ is a linear function of µ andQ. This last fact follows from the observation that xTQx =

∑i,j(xixj)qij .

Combining these observations with the assumptions that the sets X and U arenonempty and bounded, and Lemma 2.3 from Halldorsson and Tutuncu (2003), we

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have the following conclusion: Optimal values of the following pair of primal and dualproblems,

maxx∈X

{ min(µ,Q)∈U

ψλ(x, µ,Q)}, and min(µ,Q)∈U

{maxx∈X

ψλ(x, µ,Q)} (22)

are equal and are obtained at a saddle-point of the function ψλ(x, µ,Q). In otherwords, there exists a vector x ∈ X and a vector-matrix pair (µ, Q) ∈ U such that

ψλ(x, µ, Q) ≤ ψλ(x, µ, Q) ≤ ψλ(x, µ,Q), ∀ x ∈ X , (µ,Q) ∈ U , (23)

and x ∈ X , (µ, Q) ∈ U collectively solve both problems in (22).Therefore, the maximin problem (8) we formulated for the robust solution of the

asset allocation problem (2) is equivalent to finding a saddle-point of the functionψλ(x, µ,Q). This equivalence is significant because we can now use the rich literatureon saddle-point problems and in particular, the work of Halldorsson and Tutuncu(2003) where an interior-point algorithm for their solution is developed.

In a similar manner, we can develop a saddle-point formulation for the minimaxproblem (7). First, we note that the constraint minµ∈Uµ µ

Tx ≥ R can be simplified into(µL

)Tx ≥ R when Uµ is given by (4) and x ∈ X implies that x ≥ 0, which is natural

for asset allocation problems. Now, defining φ(x,Q) = xTQx and XR := {x ∈ X :(µL

)Tx ≥ R}, we obtain the following saddle-point formulation for (7): Find x ∈ XR

and Q ∈ UQ such that

φ(x, Q) ≤ φ(x, Q) ≤ φ(x, Q), ∀ x ∈ XR, Q ∈ UQ. (24)

One can deduce conditions characterizing saddle-points of the function φ(x,Q) usingthe optimality of x for minx∈XR

φ(x, Q) and of Q for maxQ∈UQφ(x, Q), which forms

the basis of the algorithm we present next.

3.3 The Algorithm

Given µ, Q, and X , the efficient frontier is the collection of portfolios that are optimalsolutions to the problem (1) for all possible values of R (or to problem (2) for all possiblevalues of λ). By representing each optimal portfolio as a point in two-dimensional spacewith coordinates equal to the standard deviation and expected return of the portfolio,one obtains the familiar depiction of this efficient frontier.

Since there are potentially infinitely many points on the efficient frontier, obtaininga precise description of this set may appear impossible. The method of critical lines inMarkowitz (1952), which is essentially a parametric quadratic programming algorithm,recognizes that there are a finite number of critical values of R in (1) with correspondingcritical optimal portfolios. As R varies between two of its consecutive critical values,say Rk and Rk+1, the optimal solutions to corresponding problems (1) can be obtainedas a convex combination of the two critical portfolios corresponding to Rk and Rk+1.Therefore, the infinite number of points on the efficient frontier can be generated by afinite algorithm, such as the method of critical lines.

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In the robust case, however, such methods are either not available or not imple-mented. While there are algorithms to solve the robust portfolio selection/asset allo-cation problems (see, Halldorsson and Tutuncu (2003), Goldfarb and Iyengar (2003)),these algorithms find a single point on the robust efficient frontier. The literatureon parametric min-max problems or parametric saddle-point problems appears to besurprisingly sparse–we are aware of only one reference, and there the emphasis is oncontinuity properties of the solutions of such problems rather than algorithms, Morganand Raucci (1997). Therefore, to generate the efficient frontier, we will first determinethe robust efficient portfolios with the lowest and highest expected returns, discretizethe range between these two extremes to obtain a finite number of set levels of theexpected return and solve the problem (7) for each set level of the expected return. Aswe discussed at the end of Section 3.2, assuming that x ∈ X , the min-max problem (7)is equivalent to the saddle-point problem (24).

To obtain the robust efficient portfolios with highest and lowest expected returnsas well as to solve the problem (7) (or, equivalently, the problem (24)) for each inter-mediate value we use the saddle-point algorithm (SP Algorithm, for short) developedby Halldorsson and Tutuncu (2003). This is an interior-point path-following methodwith computationally attractive polynomial-time convergence guarantees. We includea description of this method in the Appendix of this article.

We are now ready to formally present the algorithm that generates a discrete ap-proximation to the robust efficient frontier:

Robust Efficient Frontier Algorithm

1. Solve problem (7) without the expected return constraint using the SP Algorithm.

Let xmin denote its optimal solution. Set Rmin =(µL

)Txmin.

2. Solve problem (8) with λ = 0. Let xmax denote its optimal solution. Set Rmax =(µL

)Txmax and ∆ = Rmax −Rmin.

3. Choose K, the number of desired points on the efficient frontier. For R ∈ {Rmin+∆

K−1 , Rmin + 2 ∆K−1 , . . . , Rmin + (K − 2) ∆

K−1} solve problem (7) with the expectedreturn constraint using the SP algorithm.

As we mentioned above, in Steps 1 and 3 of the Robust Efficient Frontier Algorithm,we actually solve the saddle-point problem (24) that is equivalent to the correspondingmin-max problem (7). Step 1 generates the robust efficient portfolio that has theoverall minimum worst-case variance, without any regard for the expected returns.The worst-case expected return of this portfolio is the the minimum expected returnfor a robust efficient portfolio since no portfolio with a smaller return (and necessarilyhigher variance) would be efficient.

In contrast, Step 2 finds the robust efficient portfolio with the highest (worst-case)expected return without giving any consideration to the variance of the portfolio. Thisturns out to be a trivial problem. We want to solve the following maximin problem:

maxx∈X

{minµ∈Uµ

µTx}. (25)

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When Uµ is as in (4) and x ≥ 0 when x ∈ X , this problem is equivalent to the followingLP and can be solved easily:

maxx∈X

(µL

)Tx. (26)

Once we determined the robust efficient portfolios with the lowest and highest worst-case expected returns, in Step 3, we generate a sequence of robust efficient portfolioswhose expected returns lie in between these two extremes with equal increments.

4 Computational Results

In this section, we apply the robust asset allocation methods discussed in the previoussections to market data and compare the behavior of the solutions obtained by therobust optimization technique and the solutions obtained by standard approaches usingnominal data.

For our first experiment, we use a universe of 5 asset classes: large cap growthstocks, large cap value stocks, small cap growth stocks, small cap value stocks, andfixed income securities. To represent each asset class, we use a monthly log-return timeseries of corresponding market indices: Russell 1000 growth and value indices for largecap stocks, Russell 2000 growth and value indices for small cap stocks, and LehmanBrothers US Intermediate Government/Credit Bond index for fixed income securities.Lehman Brothers U.S. Intermediate Government/Credit Bond Index is an unmanagedindex generally representative of government and investment-grade corporate securitieswith maturities of 1-10 years. Our time series data spans the period January 1979 toJuly 2002, a total of n = 283 months.

Using this data, we computed the historical means and covariances of the five indicesmentioned above. Further, we computed lower and upper bound vectors and matriceson means and covariances using a bootstrapping strategy. Namely, a time series oflength n was chosen for each index by bootstrapping from the available observationsand means and covariances were computed for these series. This process was repeated3000 times and the quantiles of the statistics were computed. Table 1 lists the 2.5, 50,and 97.5 percentiles for means of monthly returns and covariances of these returns.

The ranges for the return and covariance estimates presented in Table 1 can beconsidered wide. However, most of these ranges are not wide enough to include thenegative or very high returns and/or negative correlations often observed in shorterobservation periods. By bootstrapping over the entire history of the time series, weobtained bounds that can be considered reliable for average behavior of returns overlong investment periods. We will discuss an alternative method to generate lower andupper bounds on returns and covariances below which may be more suitable for robustasset allocation over shorter investment horizons.

Using the data presented above, we generated the classical and robust efficientfrontiers. Figure 1 depicts the classical efficient frontier obtained by using the 50 per-centile values for expected returns and covariances as inputs as well as the robustefficient frontiers obtained by using the 2.5 and 97.5 percentile values for expectedreturns and covariances as inputs. Note that the robust efficient frontier lies below the

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10−2× Ru 1000 Gr Ru 1000 Va Ru 2000 Gr Ru 2000 Va LB IT Gov/Cre2.5 percentile 0.3398 0.6330 -0.1358 0.5866 0.586850 percentile 0.9644 1.1135 0.7221 1.1726 0.7449

97.5 percentile 1.5602 1.5825 1.5497 1.7145 0.9029

10−3× Ru 1000 Gr Ru 1000 Va Ru 2000 Gr Ru 2000 Va LB IT Gov/Cre

Ru 1000 Gr2.21472.88913.6629

Ru 1000 Va1.34931.84172.4820

1.30601.74272.3011

Ru 2000 Gr2.39283.28704.3749

1.41382.13613.0965

3.84495.15516.7911

Ru 2000 Va1.29491.92042.7833

1.12121.68792.4465

2.12453.08474.4034

1.62472.41823.5308

LB IT Gov/Cre0.04770.13460.2162

0.06280.14410.2224

-0.03320.08590.1950

0.01520.11580.2116

0.13370.18480.2500

Table 1: 2.5, 50, and 97.5 percentiles of mean monthly log-returns as well as the entries of thecovariance matrix obtained from bootstrapped samples. Only the lower diagonal entries in thecovariance matrix are listed for brevity.

classical efficient frontier since it depicts the worst-case values of the expected returnsand standard deviations. In contrast, the classical efficient frontier shows nominalvalues of these quantities. Therefore, a direct comparison of the classical and robustefficient portfolios should not be made based on this figure. They are presented to-gether to show how different the efficient frontier may look if optimization inputs areinexact/incorrect.

4 6 8 10 12 14 16 186

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16Classical and robust efficient frontiers

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Classical efficient frontier with nominal inputsRobust efficient frontier with worst−case inputs

Figure 1: The classical efficient frontier found using nominal data and the robust efficient frontierfound using worst-case data.

We mentioned that we used the 2.5 and 97.5 percentiles listed in Table 1 as the

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lower and upper bounds µL, µU , QL, and QU when computing the robust efficientportfolios. We observe that QU obtained in this manner is a positive definite matrix.Therefore, using the result of Proposition 2, the robust efficient portfolios were foundusing the classical mean-variance optimization approach with inputs µL and QU .

Next, we compare the compositions of the classical and robust efficient portfolios.They are presented in Figure 2. On the classical efficient frontier, the lowest riskefficient portfolios are obtained, as expected, using the fixed income securities. Asone moves along the efficient frontier toward the efficient portfolio with the highestexpected return, fixed income securities are gradually replaced by a mixture of large-cap and small-cap value stocks. Close to the high-return end of the frontier, large-capstocks are also phased out and one gets a portfolio consisting entirely of small cap valuestocks. In contrast to the classical efficient portfolios, robust efficient portfolios focusalmost all equity holdings in large cap value stocks. A mixture of small cap growthand value stocks are used only in very small amounts and only at the low-risk end ofthe frontier. One might be surprised that the robust efficient portfolios are even lessinclusive (i.e., contain fewer asset classes) than the classical efficient portfolios–we willcomment on this observation in our conclusion.

10 11 12 13 14 150

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Russell 1000 ValueRussell 2000 ValueLB IT Gov/Cre

7.3 7.4 7.5 7.6 7.7 7.80

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Composition of robust efficient portfolios using 2.5 & 97.5 percentile bounds

Russell 1000 ValueRussell 2000 GrowthRussell 2000 ValueLB IT Gov/Cre

Figure 2: The composition of the classical and robust efficient portfolios. 2.5 and 97.5 percentilesof means and covariances of bootstrapped samples were used to describe the uncertainty intervalsfor robust portfolios.

We then repeated the robust asset allocation algorithm on the same set of assetclasses using a different method to generate the lower and upper bounds for the inputparameters. We considered moving windows of four years and computed mean returnsand covariances in each such window. Then, we computed different quantiles of themoving window time series. Table 2 lists the 5 and 95 percentiles for means of monthlylog-returns and covariances of these returns. Interestingly, QU obtained with thismethod is also positive definite and is the unique worst-case covariance matrix withinthe interval for all possible feasible portfolios. As can be seen from these tables, this

14

process generates wider bounds on the returns and covariances than the bootstrappingstrategy discussed above. We have the following explanation for this observation. Thebootstrapping strategy we used considers (and averages out) 23 years of return dataand therefore is not able to identify significantly volatile nature of returns over shortand mid-term horizons. Moving averages with two-to-four year windows detect suchbehavior and result in wider interval estimates for the returns and covariances.

10−2× Ru 1000 Gr Ru 1000 Va Ru 2000 Gr Ru 2000 Va LB IT Gov/Cre5 percentile 0.3093 0.5333 -0.0666 0.2560 0.4570

95 percentile 2.1511 1.9658 1.7425 2.0545 1.2664

10−3× Ru 1000 Gr Ru 1000 Va Ru 2000 Gr Ru 2000 Va LB IT Gov/CreRu 1000 Gr 0.8309/5.2834Ru 1000 Va 0.5621/3.1729 0.6309/2.8234Ru 2000 Va 0.9239/5.4941 0.5957/3.8920 1.7901/8.9472Ru 2000 Gr 0.5265/3.7516 0.4963/3.2523 0.9815/5.2682 0.7193/4.4271

LB IT Gov/Cre -0.0555/0.3909 -0.0313/0.3582 -0.0737/0.3976 -0.0307/0.3819 0.0582/0.5433

Table 2: Percentiles of 4-year moving averages of monthly log-returns and covariances of monthlylog-returns.

Figure 3 illustrates the robust efficient frontier and the composition of robust effi-cient portfolios when we use the percentiles given in Table 2 as the uncertainty bounds.Efficient portfolios are very similar to those we found with the previous set of bounds,except that some of the low-risk efficient portfolios of the previous set are inefficientfor the problem with the current bounds.

6 8 10 12 14 16 18 205.5

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7Robust efficient frontier

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Composition of robust efficient portfolios using 5 & 95 percentile bounds

Russell 1000 ValueLB IT Gov/Cre

Figure 3: The efficient frontier and the composition of the efficient portfolios found using the robustasset allocation approach. 5 and 95 percentiles of 4-year moving averages means and covarianceswere used to describe the uncertainty intervals for these inputs.

We conclude this experiment by comparing the (σ, µ) frontiers for classical and ro-bust efficient portfolios under two scenarios. Since robust efficient portfolios generated

15

with two different sets of bounds were very similar, we just focus on the first set andignore the other in this comparison. First, we plot the standard deviation-expectedreturn profiles of the generated portfolios assuming that the expected returns and co-variances were actually equal to the point estimates used for the classical mean-varianceoptimization approach. Under this scenario, portfolios coming from the classical MVOapproach only slightly outperform the portfolios generated with the worst-case in mind.Next, on the same graph we plot the standard deviation-expected return profiles of theportfolios generated assuming that actual expected returns and covariances were theworst-case values within the lower and upper bounds used for robust optimization.Figure 4 shows these plots.

Compared to the nominal case, the difference in the worst-case performances ofthe two sets of efficient portfolios is greater. We observe that the performance ofclassical efficient portfolios deteriorate significantly at the high-return end with worst-case inputs. Furthermore, Figure 4 suggests that one of the most significant benefits ofthe robust approach is in risk reduction for worst-case scenarios. This is a consequenceof the (σ, µ) frontiers being relatively flat with worst-case inputs. For example, therobust efficient portfolio achieving a 7.5% worst-case annualized expected return hasan 8% standard deviation, while the classical efficient portfolio with a 7.5% worst-caseannualized expected return has approximately 12% standard deviation indicating thatit is significantly riskier.

4 6 8 10 12 14 16 18 20 227

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(σ, µ) frontiers realized with nominal and worst−case inputs

Classical efficient portfolios with nominal inputRobust efficient portfolios with nominal inputRobust efficient portfolios with worst−case inputsClassical efficient portfolios with worst−case inputs

Figure 4: (σ, µ)-profiles of classical and robust efficient portfolios when actual moments are (i)equal to their point estimates, (ii) equal to their worst possible values within given bounds.

In our second experiment, we used a wider set of asset classes: growth and valuestocks in large-cap, mid-cap, and small-cap categories, intermediate term fixed-incomesecurities, international stocks, real estate securities, and high-yield corporate bonds.To represent the domestic equity classes, we used the log-return time series for Wilshire

16

Target indices corresponding to each category. We use the Lehman Brothers Interme-diate Government/Credit index, MSCI EAFE (Europe, Australasia, Far East) index,Wilshire Real Estate Securities index, and Lehman Brothers High-Yield Bond indexfor the remaining categories. Our time series data spans the period July 1983 to July2002, a total of n = 229 months.

Using this data, we computed the historical means and covariances of the log-returns of the ten indices mentioned above. Further, we computed lower and upperbound vectors and matrices on means and covariances using the 4-year moving averagesas was done in the first experiment. Table 3 lists the 5, 50, and 95 quantiles for meansof monthly log-returns and covariances of these returns.

10−2× Ws LC Gr Ws LC Va Ws MC Gr Ws MC Va Ws SC Gr Ws SC Va LB IT G/C MS EAFE Ws RE Sec LB Hi-Yld5 % 0.5711 0.2586 0.0314 0.3622 0.0352 0.2076 0.4534 -0.2109 -0.3670 0.0644

50 % 1.2486 1.0909 1.0480 1.1863 0.8681 1.1460 0.6251 0.7180 0.6537 0.860395 % 2.3590 1.8205 1.5684 1.6851 1.5460 1.7965 0.9702 2.9500 1.2550 1.2857

10−3× Ws LC Gr Ws LC Va Ws MC Gr Ws MC Va Ws SC Gr Ws SC Va LB IT G/C MS EAFE Ws RE Sec LB Hi-Yld

Ws LC Gr0.76332.26164.2952

Ws LC Va0.41181.56532.7668

0.67431.55862.5193

Ws MC Gr0.84512.42854.3482

0.51591.82213.0201

1.52233.17785.4658

Ws MC Va0.37761.38102.7408

0.52211.41952.4136

0.57811.98123.3073

0.56981.58513.1725

Ws SC Gr0.86662.62794.7417

0.51801.75033.2677

1.67093.51125.9247

0.59662.03553.6341

1.93424.05166.6948

Ws SC Va0.35541.21842.5611

0.43641.24281.9703

0.58421.87453.1768

0.49691.42112.3507

0.62842.01043.5407

0.47841.37952.2155

LB IT G/C-0.06800.11800.2240

-0.06160.15250.2006

-0.09020.06230.1925

-0.02500.12500.1801

-0.05840.03360.1761

-0.00700.09770.1516

0.05770.09240.1618

MS EAFE0.16371.30352.5973

0.30851.02021.4259

0.43741.51542.7885

0.25420.90731.3379

0.50271.57562.6630

0.22750.84341.3036

-0.08560.05850.2000

1.25002.36824.0717

Ws RE Sec0.18610.70612.3900

0.31661.04851.8004

0.41881.33763.0029

0.35981.26462.0014

0.54841.39053.4410

0.39751.17631.9695

-0.02130.06720.0979

0.24330.56871.3112

0.95261.66912.3312

LB Hi-Yield0.11310.59100.9148

0.15200.42890.7006

0.15660.69011.1368

0.13760.46580.7611

0.16410.74781.2796

0.13360.46350.7498

0.00620.06330.1519

0.13150.30220.5425

0.11850.32640.8800

0.13450.30880.8082

Table 3: 5, 50, and 95 percentiles of 4-year moving averages of monthly log-returns and covariancesof monthly log-returns.

In Figures 5 and 6 we illustrate the efficient frontiers and the composition of theefficient portfolios found using the classical and robust asset allocation approaches1.The composition graphs illustrate the similarities as well as the stark differences be-tween the choices made by the two approaches: Both approaches invest a substantialproportion of their portfolios in fixed income securities at the low-risk end of theirspectrum and gradually phase them out of the portfolios as one moves toward the high

150 percentile values of the covariances results in a matrix with a negative eigenvalue. Since this eigenvalueis very small in absolute value, we simply modified the 50 percentile matrix with a small multiple of theidentity to obtain a legitimate (positive semidefinite) covariance matrix. This modified matrix was used todetermine the classical efficient portfolios.

17

return end of the frontier. High yield bonds that constitute more than 60% of themid-range portfolios in the nominal case are too risky for the robust approach and arereplaced by government bonds. For the equity portion of the portfolios, the classicalapproach chooses a mixture of large cap growth stocks as well as mid and small-capvalue stocks. In contrast, the robust approach focuses on the large-cap growth stocksand includes small amounts of real estate securities at the low-risk end of the frontier.Once again, we observe that the robust portfolios are less inclusive.

0 5 10 15 20 254

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Classical efficient frontier with nominal inputsRobust efficient frontier with worst−case inputs

Figure 5: The classical efficient frontier found using nominal data and the robust efficient fron-tier found using worst-case data. 5 and 95 percentiles of 4-year moving averages for means andcovariances were used to describe the “worst-case inputs”.

Next, we examine the effect of the length of the moving window used for the compu-tation of lower and upper bounds describing uncertainty intervals. Instead of a 4-yearwindow, we use 2-year windows and compute the corresponding percentiles. Using 5and 95 percentiles as the lower and upper bounds leads to uncertainty sets that aretoo wide to be of practical interest. In fact, with such bounds, the robust efficientfrontier degenerates into a very small set centered at the pure fixed-income portfolio.When 10 and 90 percentiles of 2-year moving averages are used, we obtain boundsthat are similar to those found with 5 and 95 percentiles of 4-year moving averagesand virtually identical robust efficient portfolios. We also used 25 and 75 percentilesof the 2-year moving averages to describe our uncertainty sets. This choice leads tointervals that are often, but not always, tighter than those we found with 5 and 95percentiles of 4-year moving averages. We list the lower and upper bounds obtainedin this manner in Table 4 and illustrate the corresponding robust efficient frontier aswell as the composition of robust portfolios in Figure 7. As opposed to the previousexperiments, the upper bound matrix QU obtained in this manner is not a positivesemidefinite matrix, and therefore the robust efficient portfolios were generated using

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9 10 11 12 13 14 15 160

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5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 70

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Wilshire LC GrowthLB IT Gov/CreWilshire Real Est Sec

Figure 6: The composition of the classical and robust efficient portfolios. 5 and 95 percentiles of4-year moving averages for means and covariances were used to describe the uncertainty intervalsfor robust portfolios.

the algorithm presented in Section 3.3. The resulting efficient portfolios are differentfrom the previous two sets, and mostly focus on large-cap value stocks for their equityallocations.

10−2× Ws LC Gr Ws LC Va Ws MC Gr Ws MC Va Ws SC Gr Ws SC Va LB IT G/C MS EAFE Ws RE Sec LB Hi-Yld25 % 0.5952 0.6220 0.3428 0.5809 0.2408 0.4752 0.5063 0.3186 0.2384 0.388075 % 2.0878 1.6486 1.5027 1.7666 1.4305 1.8257 0.8516 1.4490 1.3859 1.1550

10−3× Ws LC Gr Ws LC Va Ws MC Gr Ws MC Va Ws SC Gr Ws SC Va LB IT G/C MS EAFE Ws RE Sec LB Hi-Yld

Ws LC Gr1.44213.5834

Ws LC Va0.68792.1060

0.86532.5030

Ws MC Gr1.40913.5458

0.77222.6544

1.82394.7440

Ws MC Va0.62332.0468

0.67482.3619

0.86723.0221

0.66722.6682

Ws SC Gr1.39573.7728

0.65472.8021

1.99845.0969

0.69133.0908

2.32775.8410

Ws SC Va0.55881.8277

0.55271.7764

0.83552.7839

0.59822.1986

0.79962.9850

0.57632.0594

LB IT G/C0.02380.1935

0.04650.1876

-0.03000.1595

0.05140.1700

-0.04140.1540

0.03890.1286

0.06420.1203

MS EAFE0.36022.1599

0.34901.5159

0.57752.5945

0.32651.4203

0.63132.5283

0.28831.2517

-0.02850.1702

1.85142.8549

Ws RE Sec0.24201.3167

0.45581.5972

0.54711.9951

0.53781.8447

0.63852.2476

0.49301.5504

-0.00250.0932

0.26621.0387

0.82112.0566

LB Hi-Yield0.24440.7214

0.17860.5904

0.24980.8910

0.15970.5887

0.25250.9822

0.15700.5634

0.02240.0933

0.11470.4623

0.14670.4919

0.17930.4389

Table 4: 25 and 75 percentiles of 2-year moving averages of monthly log-returns and covariancesof monthly log-returns.

We now compare the performance of the portfolios generated by the classical androbust asset allocation approaches under three scenarios. In Figures 8 and 9 we plot the

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Wilshire LC GrowthWilshire LC ValueLB IT Gov/CreLB Hi−Yld

Figure 7: The efficient frontier and the composition of the efficient portfolios found using therobust asset allocation approach. 25 and 75 percentiles of 2-year moving averages for means andcovariances were used to describe the uncertainty intervals for these inputs.

standard deviation-expected return profiles of the generated portfolios assuming thatthe realized returns and covariances were actually equal to the point estimates usedfor the classical mean-variance optimization approach. Under this scenario, portfolioscoming from the classical MVO approach outperform the portfolios generated with theworst-case in mind.

In Figures 8 and 9, we also plot the standard deviation-expected return profiles ofthe generated portfolios assuming that the realized returns and covariances were amongthe worst possible from the uncertainty sets described in Tables 3 and 4, respectively.Note that the worst-case realization of the data depends on the particular portfolio onehas (except when QU is positive semidefinite, see Proposition 2) and therefore, there isno unique realization of the uncertain parameters that is uniformly bad for all portfoliosthat we could use for these graphs. For the graphs in Figures 8 and 9, we used theworst case realization of the data for the robust minimum variance portfolio. As is clearfrom the graph, portfolios generated to perform optimally in the worst-case scenario,significantly outperform the portfolios generated using the classical MVO approachwithout considering data uncertainty. Portfolios generated using bounds derived from4-year moving windows perform reasonably well under the two scenarios favoring theother portfolio sets and they perform much better than other portfolio sets under thescenario it was chosen to optimize over. As such, these portfolios provide a sensiblealternative to classical efficient portfolios that suffer greatly when the input estimatesare inaccurate.

As a final exercise, we test the stability of the efficient portfolios generated by theclassical and robust approaches through time. For this purpose, we first consider theportion of our data that was available at the end of 1997. With this data, using theapproaches outlined above, we generate mean and covariance estimates for classical

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18(σ, µ) frontiers realized with nominal and worst−case inputs from 4−year moving averages

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Classical efficient portfoliosRobust efficient portfolios (4 yr.)Robust efficient portfolios (2 yr.)

Figure 8: (σ, µ)-profiles of classical and robust efficient portfolios when actual moments are (i)equal to their point estimates, (ii) equal to their worst possible values within bounds found using4-year moving windows.

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Classical efficient portfoliosRobust efficient portfolios (4 yr.)Robust efficient portfolios (2 yr.)

Figure 9: (σ, µ)-profiles of classical and robust efficient portfolios when actual moments are (i)equal to their point estimates, (ii) equal to their worst possible values within bounds found using2-year moving windows.

mean-variance optimization and use 4-year moving averages to generate lower andupper bounds for robust optimization. In addition to identifying the classical and

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robust efficient frontiers, we also compute the portfolios that maximize the Sharpe ratioin the classical and robust senses. Then, we repeat this process with data availablethrough the end of each year between 1998 and 2001. Each additional year of datachanges the mean and covariance estimates, and we track the effects of these changes onefficient portfolio compositions. While the compositions of classical efficient portfoliosvary significantly from year to year, robust efficient portfolios have remarkably smallturnover ratios indicating that investors following such strategies will incur minimaltrading costs at rebalancing dates. Figures 10 and 11 illustrate our arguments. Thefirst pair of figures show how the composition of the portfolios that maximize theSharpe ratio using classical and robust approaches change over the course of 5 years.The second pair of figures depict the composition of the classical and robust efficientportfolios that are at the “middle” of the efficient frontier (representing, approximately13.5% annual return in the nominal case and 7.7% annual return in the worst-case).

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Figure 10: The composition of robust portfolios maximizing the Sharpe ratio remains relativelyconstant through a 5 year period.

5 Comments and Conclusion

Building on recent research in robust optimization, this article presents a novel ap-proach to asset allocation problems under data uncertainty. As opposed to the clas-sical approach, where one estimates the inputs (returns and covariances) to the assetallocation problem and then treats them as certain and accurate, the approach pre-sented here advocates the description of input estimates in the form of uncertainty sets.Robust asset allocation refers to finding an asset allocation strategy whose behaviorunder the worst possible realizations of the uncertain inputs is optimized. The articlepresents an algorithm for generating robust efficient portfolios using a variant of aninterior-point method for saddle-point problems, and discusses an implementation ofthis algorithm.

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Composition of mid−range efficient portfolios (nominal input)

Wilshire LC GrowthWilshire LC ValueWilshire MC ValueWilshire SC ValueMSCI EAFELB Hi−YldLB IT Gov/Cre

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Composition of mid−range robust efficient portfolios

Wilshire LC GrowthWilshire LC ValueLB IT Gov/Cre

Figure 11: The composition of mid-range robust efficient portfolios remain relatively constantwhile corresponding classical efficient portfolios experience large turnovers through a 5 year period.

Our analysis demonstrates some key properties of asset mixes formed using robustoptimization techniques: (i) significantly better worst-case behavior, (ii) stability overtime, and (iii) concentration on a small set of asset classes. While the first two prop-erties are expected, the third one appears surprising. We now comment on all threeproperties.

The numerical experiments with the algorithm presented in Section 4 suggest thatthe worst-case behavior of portfolios of different asset classes can be improved signifi-cantly using the robust asset allocation approach, often with small performance losseson more likely scenarios. As the size of uncertainty sets increases, both the benefits ofrobust portfolios under worst-case scenarios and their under-performance under mostlikely-scenarios appear to increase as well. This trade-off suggests that a cost-benefitanalysis of the size of uncertainty sets needs to be performed and the right size willdepend on the risk-attitudes of individual investors.

Another property of robust efficient portfolios demonstrated by our results is thatthey remain relatively unchanged over long periods of time. As illustrated in Figures 10and 11, recomputing robust efficient portfolios as new data becomes available generatesremarkably small turnover ratios and, therefore, such portfolios would experience onlymodest trading costs if they are rebalanced regularly. Moreover, since robust optimalportfolios computed at the beginning of a long investment horizon appear to remainoptimal or near optimal throughout this horizon, they represent attractive alternativesfor buy-and-hold investors.

An observation (and sometimes, complaint) expressed by users of the classical assetallocation methods/software is that the portfolios generated often concentrate on asmall number (say, 2 or 3) of the asset classes available for investment. This appearscontradictory to the well-known benefits of diversification. We now give a heuristicexplanation of this phenomenon that also helps us interpret the behavior of robust assetallocation methods. Trying to negotiate the two competing objectives of maximizing

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returns and minimizing risk, the MVO algorithm, in a way, generates two rankingsof the assets (or asset classes)–one for high return potentials and one for low riskpotentials. Highest return portfolios are obtained by picking one or a few of the assetstopping the high-return list. As the algorithm moves from the high return end of theefficient frontier toward the low risk end, these assets are gradually replaced by thosethat top the low-risk list. Assets dominated by another asset in both lists almost neverfind a place in an efficient portfolio.

While one may expect the robust efficient portfolios to be “more diversified” ormore inclusive than their classical counterparts–this comparison may be hard to makewithout an appropriate metric–we observed a similar phenomenon to the one describedin the previous paragraph in our computational experiments with the robust assetallocation approach. This time, it appears, the method ranks the worst-case returnpotentials of the assets as well as their worst-case riskiness, and then proceeds in asimilar manner as before, from the assets with highest worst-case returns to those withlowest worst-case variance. This behavior is apparent from Table 3 and Figures 5,6. Instead of the small-cap value stocks (whose 50 percentile returns are the highest)that we see at the high-return end of the classical efficient portfolios, large-cap growthstocks, whose worst-case returns are highest, are seen at the high-return end of therobust efficient portfolio.

Overall, we conclude that by directly addressing some of the weaknesses of classicalMVO, robust optimization provides a valuable asset allocation vehicle to conservativeinvestors.

ACKNOWLEDGMENT

The first author would like to thank L. N. Vicente for discussions related to thematerial in Section 2.2. We also thank three anonymous referees for useful commentsand suggestions. The first author was supported in part by NSF under Grant No.CCR-9875559 and Grant No. DMS-0139911.

References

[1] Bawa, V. S., S. J. Brown, and R. W. Klein. (1979). Estimation Risk and OptimalPortfolio Choice, North-Holland, Amsterdam, Netherlands.

[2] Ben-Tal, A. and A. Nemirovski. (1999). “Robust Solutions of Uncertain LinearPrograms,” Operations Research Letters 25, 1–13.

[3] Goldfarb, D. and G. Iyengar. (2003). “Robust Portfolio Selection Problems,”Mathematics of Operations Research 28 (1), 1–38.

[4] Halldorsson, B. V. and R. H. Tutuncu. (2003). “An interior-point method for aclass of saddle-point problems,” Journal of Optimization Theory and Applications116(3), 559–590.

[5] Jen, E. (2001). Working Definitions of Robustness,” SFI Robustness Sitehttp://discuss.santafe.edu/robustness, RS-2001-009.

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[6] Lobo, M. S. and S. Boyd. (1999). “The Worst-Case Risk of a Portfolio,” Informa-tions Systems Laboratory, Stanford University.

[7] Markowitz, H. (1952). “Portfolio Selection,” Journal of Finance 7, 77–91.

[8] Michaud, R. O. (1989). “The Markowitz Optimization Enigma: Is OptimizedOptimal?”, Financial Analysts Journal 45, 31–42.

[9] Michaud, R. O. (1998). Efficient Asset Management, Harvard Business SchoolPress, Boston, Massachusetts.

[10] Morgan, J., and R. Raucci. (1997). “Continuity properties of e-solutions for gen-eralized parametric saddle point problems and application to hierarchical games,”J. Math. Anal. Appl. 211, 30–48.

[11] Nesterov, Yu. and A. Nemirovskii. (1994). Interior-Point Polynomial Algorithmsin Convex Programming, SIAM, Philadelphia, Pennsylvania.

6 Appendix

Robust Efficient Frontier Algorithm we presented in Section 3.3 uses a saddle-pointalgorithm in each intermediate step. This saddle-point algorithm, originally developedin Halldorsson and Tutuncu (2003), is from the class of interior-point path-followingalgorithms. Such methods are based on the concept of a central path in the relativeinterior of the feasible set of points that converges to a saddle-point of the given functionwithin this set (or, to an optimal point in the case of optimization problems). Iteratesare generated in close proximity to this path (hence the name path-following) so thatthey can be guaranteed to converge to a saddle-point, just like the central path.

Let us describe the central path in more detail. First, we define the followingsaddle-barrier function:

φt(x,Q) := tφ(x,Q) + F (x)−G(Q),

where F (x) and G(Q) are self-concordant barrier functions (see, Nesterov and Ne-mirovskii (1994)) for the relative interiors X 0

R and U0Q of the sets XR and UQ:

F (x) := −n∑

j=1

log(xj)− log(µTLx−R), x ∈ X 0

R

G(Q) := −∑

1≤i≤j≤n

log(QUij −Qij)−

∑1≤i≤j≤n

log(Qij −QLij)− log det(Q), Q ∈ U0

Q.

As long as X 0R and U0

Q are nonempty, there is a unique saddle-point of the function φt

for each t ≥ 0. These saddle-points form a continuous trajectory in the interior of theset of feasible points called the central path of the saddle-point problem (24). As t tendsto ∞, saddle-points of the functions φt, and therefore, the central path converge to asaddle-point of the problem (24). Our algorithm generates iterates that are in closeproximity to the saddle-points of functions φt for increasing values of t and convergeto a solution using this strategy.

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The algorithm we present below starts from a trial point (x0, Q0) close to a pointon the central path corresponding t = t0 and using updates generated by Newton’smethod, determines iterates that are close to points on the central path correspondingto exponentially increasing values of t. Proximity of points to the central path ismeasured by the following function, called the Newton decrement:

η(φt, x,Q) :=√η2(φt

Q, x) + η2(−φtx, Q), with,

η(ψ, z) :=√∇ψT (z)[∇2ψ(z)]−1∇ψ(z),

φtQ(x) := tφ(x,Q) + F (x), and

φtx(Q) := tφ(x,Q)−G(Q).

The measure η(φt, x,Q) is always nonnegative and is zero only on central points. Fur-thermore, a small value of the Newton decrement function implies close proximity tothe central path. By a judicious choice of the parameters, the algorithm makes surethat the Newton decrement is small for all iterates generated, guaranteeing eventualconvergence. We are now ready to present the saddle-point algorithm formally:

Saddle-Point Algorithm (SP Algorithm)

1. Initialization:Choose α > 0 and β > 0. Find a t0 > 0 and (x0, Q0) ∈ X 0

R × U0Q that satisfies

η(φt0 , x0, Q0) ≤ β. Set k = 0.

2. Iteration:while tk < MSet

tk+1 = (1 + α)tk. (27)

Take a full Newton step:

(xk+1, Qk+1) = (xk, Qk)−[∇2φtk+1

(xk, Qk)]−1

∇φtk+1(xk, Qk). (28)

Set k = k + 1.end

The parameters α and β need to satisfy certain conditions as prescribed in Halldorssonand Tutuncu (2003) to ensure that

η(φtk , xk, Qk) ≤ β

for all k. This implies that all iterates are close to the central path. The equation (27)indicates that tk is growing exponentially. Therefore, it eventually exceeds M , which isa large positive number chosen in a way to ensure that the final iterate is close enoughto the actual saddle-point. We leave out the remainder of the implementation detailsfor the sake of brevity. The analysis presented in Halldorsson and Tutuncu (2003) canbe adapted to problem (24) to show the polynomiality of this saddle-point algorithm.

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